An improved error estimate for reduced-order models of ... - IEEE Xplore

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TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 35, NO. 3. MARCH 1990 P. L. Pio, “Euler angle transformations.” I€€€ Trans. Automat. Contr., vol. AC-11. pp. 7077715. Oct. 1960 H. Goldstein. Classical Mechanics. 2nd ed. Reading, MA: Addison-Wesley, 1980. E. T. Whittaker. A Treatise on the Analytical Dynamics ojParticles and Rigid Bodies, 4th ed. Cambridge, England: Cambridge University Press, 1964. T. F. Wiener, “Theoretical analysis of gimballess inertial reference equipment using delta-modulated instruments,” Doctoral dissertation. M.I.T., Cambridge, MA, 1962. J. R. Wertz, ed.. Spacecrajt Attitude Determination and Control. Dordrecht, The Netherlands: Reidel, 1978. M. 0. Rodrigues. “Des lois gCometriques qui regissent les deplacement d’un systeme solide dans I’espace. et de la variation des coordonnees provenant de ces deplacements consideres independamment des causes qui peuvent Ies produire,” J . de Mathematiques, Pures et Appliquees, vo1.5. pp. 380-440, 1840. R. E. Roberson, “Kinematic equations for bodies whose rotation is described by the Euler-Rodrigues parameters,” AIAA J . , vol. 6 , pp. 916-917, Jan. 1968. J . W. Gibbs, Scientific Papers, vol. 11. New York: Dover, 1961, p. 65. J . Stuelpnagel. “On the parametrization of the three-dimensional rotation group.” SIAM Rev., vol. 6, pp. 422-430, Oct. 1964. C . G . J . Jacobi. “Bemerkungen zu Einer Abhandlung Euler’s uber die Orthogonale Substitution,” in C . G . J . Jacobi’s Gesammelte Werke, 2nd e d . . vol. 111. New York: Chelsea, 1969. pp. 601-609. L. Euler, “Problema algebraicum ob affectiones prorsus singulares memorabile.” Novi Comm. Acad. Sri. Petrop.. vol. 15, pp. 75-126, 1770. J. L. Lagrange, Mecanique Analytique. 1st ed., part 11, Section VI: Sur la Rotation des Corpes. Chez La Veuve de Saint, Libraire. Rue du Fuin S. Jacques, 1788, p. 353. A. Cayley. “On the motion of rotation of a solid body.” Cambridge Math. J . , vol. Ill. pp. 224-232, 1843: also in The Collected Mathematical Papers of Arthur Cayley. Vol. I . The Cambridge University Press. 1889. New York: Johnson Reprint, 1963, pp. 28-35. A. Cayley. “On certain results relating to quarternions,” Cambridge Math. J . , vol. 111, pp. 141-145, 1843; also in The Collected Mathematical Papers of Arthur Cayley, Vol. 1, The Cambridge University Press, 1889. New York: Johnson Reprint, 1963, pp. 123-126. L. Euler, “Nova methodus motum corporum rigidorum determinandi,“ Novi Comm. Acad. Sci. Petrop., vol. 20, pp. 208-238, 1775. L. Euler, “Formulae generales pro translatione quacunque corporum rigidorum.” Novi Acad. Sci. Petrop.. vol. 20. pp. 189-207, 1775. P. Lancaster and M. Tismenetsky, The Theory ojMatrices, 2nd ed. Orlando, FL: Academic, 1985, p. 219. J . R. Westlake. A Handbwk o j Numerical Matrix Inversion and Solution of Linear Equations. New York: Wiley, 1968.

have a joint diagonal solution C a compatible way, 0

.2O

],A=[:;;

> 0 (positive definite). Partitioning, in

3, B =

where C , , A l l E K r X r ,r order model i(t

[z;]

, C =IC,,

C21

(1.4)

< n one obtains an r-dimensional reduced-

+ 1) = A l 1 i ( t )+ B I u ( t ) , y ( t ) = C l i ( t )

( 15 )

of (1.1). The associated transfer functions are G(z) =C(ZZ, - A ) - ’ B , G(z)=CI(ZZ, - A I I ) - I B I . Their difference

E ( z ) = G ( z )- G ( z )

( 1.6)

describes the deviation of the input-output behavior of the reduced-order model (1.5) from that of the original system (1.1). If

C = U 1 I m , 9 ” ’ %uo,Zm,

( 1.7)

and r = m ,t. .+mk,k < 1. AI-Saggaf [I] (see also [2]) has established the following bound on the H , -norm of the error function E: I

IIEIIH===

og;Tl I I E ( ~ ” )5I I2

mIuI.

j=k+l

In the continuous-time case, Glover [4] has shown that a stronger estimate is valid

IlG -GllH= =maxIIG(iu)-G(iu)ll w

5 2 x 0 , .

(1.8)

€R j=k-l

Improved Error Estimate for Reduced-Order Models of Discrete-Time Systems D. HINRICHSEN

AND

A. J. PRITCHARD

In [ 11 it has been pointed out that a transfer of this result to the discretetime case via the standard bilinear transformation is not possible since the bilinear transformation introduces a direct input-output coupling into the reduced-order discrete-time model. In this note we will present an elementary direct proof of the stricter estimate for the discrete-time case under substantially weaker conditions. Following the same lines, a simple proof of Glover’s result (1.8) for continuous-time systems can be given.

Abstract-In this note we derive, under weaker conditions, a discretetime counterpart of Glover’s error estimate for reduced-order models.

I. INTRODUCTION We consider finite-dimensional time-invariant discrete-time systems of the form x(t

+ 1) = A x ( t ) + B u ( t ) Y(0=

P 5 Q(P < Q ) H x * P x 5 x * Q x ( x * P x < x * Q x ) (1.1)

where ( A , B , C ) E K n X ”x K n X mx K p X ” ,K = R or C and u ( A ) C D l , the unit disk in C . If ( A , B , C) is reachable, observable, and balanced, the two Lyapunov equations C = -BB*

( 1 .2)

A * G 1 - C = -C’C

( 1.3)

A M *

~

11. ERROR ESTIMATE We will not need all the hypotheses made in [1]-[4] in our proof. In particular, we do not assume that ( A , B , C) is reachable, observable, and balanced. Suppose that 5 denotes the usual order relation between Hermitian matrices

Manuscript received October 3 , 1988. D . Hinrichsen is with the Department of Mathematics, University of Bremen, Bremen, FRG. A. J . Pritchard is with the Institute of Mathematics, University of Warwick, Coventry England. IEEE Log Number 8933220.

for all x E C ” , x

# 0.

Our analysis will proceed via the two Lyapunov inequalities ACA*-C
0 with C, diagonal, then the truncated system is asymptotically stable and one obtains an error estimate of the type (1.8). The proof of the latter result proceeds- as in the continuous-time case- by successive truncation of the last n 4 , n q - , , ’ . , n l state components. However, in contrast to the continuous-time case, the reduced-order models are not necessarily bal-

0018-9286/90/0300-0317$01.OO 0 1990 IEEE

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anced. This complicates the induction procedure and is the reason why we avoid balancing assumptions at the outset. The following lemma shows that the truncated systems will always be asymptotically stable if one of the Lyapunov inequalities (2.1) or (2.2) admits a positive definite block-diagonal solution. Lemma 2.1: Suppose that A , B , C, C are partitioned as in (1.4) and satisfy (2.2). If o ( A ) c D , and C, > 0, C, > 0, then o ( A l l ) c D , . Proof: By (2.2) and (1.4) we have

-c;c,.

A:,C,A,I +AzflC2A21 -1,5

(2.8)

A(e"))* (2.9) - A ( e " ) ) (2.10)

(2.3)

Now assume

for some X E C', X # 0, and A E C.Multiplication of (2.3) on the left by X* and on the right by X yields (1AI2

-

l)X*CIX + X * A ~ l C 2 A 2 15X

-

+ e " ~ ( e ' e I -, A ) *

< (e"I, - A)Ce-" + e"C(e"I, - A ) *

11CIXl12.

Since C, > 0 and C2 > 0, this implies ( A / 5 1. Moreover, if then

1x1 =

I,

-

C+ACA*

-

BB*

we have

A 2 1 x= 0 and C,X = 0 and hence

A

KI

All = [A2l

X XI = A

KI

In particular, the (2.2) blocks of both sides satisfy

+ ( e i B I n - -A(e'e))--IB(e'o)B*(eie)(elol,-, r -A(e"))-'*

'

C It follows that o ( A l , ) C D I if u ( A ) C D I . The next lemma is well known in the literature; see [I], [3]. We outline a proof.

Lemma 2.2: I f A , B , C are partitioned as in (1.4) and E ( z )is defined by (1.6), then E(z) = C(z)(z1,-,

-A(z))rlB(z)

5 C2(efoI,-,- A(els) ) - I * e-,' t e i e ( e " I , - , -A(e"))-'Cz,

8 ER;

(2.1 I )

see (2.5), (2.6). (Note that e"@u(AII) for all 8 t R , by Lemma 2.1.) Multiplying(2.1 I ) on the left by - A ( & ) ) * and on the right by (e"I,-, -A(e"))* yields (2.9). Equation (2.10) is obtained analogously.

(2.4)

Lemma 2.4: Under the assumptions of Lemma 2.3, if C, =

where

/3 > 0, then

Proof: Multiplying (2.9) on the left by C ( e " ) ( ~ ' Z , - , -A(e"))-I Proof: The proof depends on computing the inverse (zZ, - A)-' . and on the right by ( c ( & e ) ( & o ~ ~ - , 11-1 )* we obtain For any z E C \ o ( A ) there exist matrices X , Y , Z , T of size r x r ,

(2.6)

Lemma 2.3: Suppose that o ( A ) c D , and AEA* - C 5 - B B *

(1 5 20.

i l C ( e ~ o ) ( e ~ o l -A(e'e))-'*C*(elo) fl-r (2.7)

This, together with (2.13) proves (2.12).

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Remark 2.5: In the continuous-time case if ( A , B , C) is balanced with a simple smallest singular value U and if the corresponding state component is truncated, the HX-norm of the reduction error is exactly 20. It was pointed out in [2] that an analogous result does not hold in the discrete-time case. In fact, it follows from (2.11) that E(e")E*(e") 5 P[C(e" )(e"[, -, - A(e"))-'

+ ef'C(e'8) ( e J o I , -

A(ez'I)-' E*(@'

-

-r

C = Cz. Hence, if C = d i a g ( P I , , . . , P q )satisfies (2.1) and (2.2), we obtain the following estimate for the Hn"-norm of I( G ( z ) / /

C*(efB )e-''

*

I

As the proof shows, the theorem can also be applied if C , is void, i.e.,

C(e")C*(e")1.

I l G l l ~ x5 2 ( P i

(2.15)

Hence, strict inequality holds in (2.13) [and consequently in (2.12)] if C(efR)C*(els)> 0. c We are now in a position to prove the discrete-time counterpart of Glover's result, under weaker assumptions. Theorem 2.6: Suppose that A , B , C, C are partitioned as in (1.4) and satisfy the Lyapunov inequalities (2.1). (2.2). If u ( A ) c D , , C, > 0. and CZ = P I I n ,@ . . . @POIn, with PI,...,& > 0, then (2.16)

Proof: We proceed recursively and begin with the partition

+".+

(2.17)

Pq).

Note that in order to apply Theorem 2.6 we only require that the two Lyapunov inequalities (2.1), (2.2) have a joint solution E, so the error estimate (2.16) applies to a much wider class of reduced-order models than those based on balancing. In the balanced case we obtain the following discrete-time counterpart of Glover's error estimate (1.8). Corollary 2.7: Suppose that the discrete-time system ( I . 1) is asymptotically stable, reachable, initially observable, and balanced with Gramian Cb = f l I I , r z ,

@

' ' '

3U/zml,

01

>0 2 >

.

' '

> U / > 0. (2.18)

+ . . . +m, statecomponents

I f r = m l +...+mk,i.e.,thelastmk,l are truncated, then

IIE/~H= < 2 ( u k + l +'--+a!). corresponding to the partition C =

q =cl BPII", 9

81

< where

cGPq-lI"q..,,

" '

E;

E. Proof: C = Cb satisfies (2,1), (2.2) with equality. But even in the balanced case the greater flexibility of Theorem 2.6 can be useful. Suppose that 2: is a solution of (2. I ) , (2.2) as in Theorem 2.6. Then

=PqInu

Define

where I is the identity matrix of appropriate dimension. By Lemma 2.4 llE4(z)llHx 5 20,.

+ P , A ~ , A ~-;E: 5

-q

A;;C;Ay, + P q A ; ; A 2 1

5

- BYBY'

-c;*c;.

Hence, (Ayl, By, C y ) , CY satisfy again the assumptions of Lemma 2.4 so that we can repeat the argument. Let Ah

c4-1 1

= Cl f3PlI,, Ct?

= " '

Y

k =O

k -0

so that the diagonal entries of & will be larger than the corresponding diagonal elements of . If the multiplicities of the P ' s are the same as those of the U ' S , the error estimate (2.16) cannot be better than (2.18). However, it may be that some of the multiplicities of the P's are larger than those of the U ' S and then (2.16) can give a tighter bound for the error than (2.19). This is illustrated in the following example. Example 2.8: Consider the asymptotically stable, reachable, initially

By Lemma 2.1, u ( A 7 , )C D l . Moreover,

correspond to the partition

w

G ' ( z ) = C ~ ( Z-I Ayl)-'B7

E'(z) = G ( z )- G'(z),

A;' CYA;,'

CQ-l

2

-.36

0

0

0 0

-6.6

36

.78

0

0

0 0

-54

0

0

0

1 0

0

0

0

.25

0

0

=

qP1 C€ g-' where

@Pq--ZIn,-2,

-.05

.

5

0

) -Ayl-') Eq-I(z) = G q ( z )- G 4 - ' ( z ) , G q P 1 ( z=Cy-'(zI

Then, by Lemma 2.4,

C b = I606 4:-

'!I-'

O

O1

.

The singular values are (45.7, 15.8,4/3,16115). The Lyapunov inequalities (2.1), (2.2) hold for C=diag(45.7, 15.8,Pl . & ) providing

IIEq-I

llH2

5 2Pq-I.

PI

Continuing in this way we have 52P,,

If

E'(z)= G ' + ' ( z ) - C ' ( Z ) ,

j =q

-

1,. . . , 1

2413

02 2 16/15.

',I

j-q,q-1;

where

and

1

c0

=Pq--lIn,-,

and define

"l=

(2.19)

PI

= Pz = 413, we obtain the error estimate

IIEIIH= I 2 . 4 / 3 whereas the Glover estimate (2.18) is

l l E l l ~ =5 2(4/3

+ 16/15).

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 35, NO. 3 . MARCH 1990

REFERENCES U. M. AI-Saggaf, “On model reduction and control of discrete time systems,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1986. U. M. AI-Saggaf and G. F . Franklin, “An error bound for a discrete reduced order model of a linear multivariable system,” IEEE Trans. Automat. Contr.. vol. AC-32, Sept. 1987. D. F. Enns. “Model reduction with balanced realizations: An error hound and a frequency weighted generalization,” in Prm. 23rd CDC. Las Vegas, NV, 1984. K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L‘=-error hounds.” Int. J. Contr., vol. 39. no. 2, Apr. 1984. L. Pernebo and L. M. Silverman, “Model reduction via balanced state space representations,” IEEE Trans. Automat. Contr., vol. AC-27. pp. 382-387. 1982.

On the Polynomial Equations for the MIMO LQ Stochastic Regulator E. MOSCA, L. GIARRE’,

AND

A. CASAVOLA

Absfracf-In this note, the role played by three polynomial equations in the LQ stochastic regulation problem is discussed. The emphasis is on establishing conditions under which the LQ regulator can be obtained via the solution of a single uncoupled Diophantine equation.

Consequently, the latter equation is sometimes referred to as the implied equation [2]. In [2], conditions were established under which the minimum degree solution w.r.t. Y of (3) yields the same polynomial matrices X and Y as the ones given by the minimum degree solution w.r.t. Z of ( I ) and (2). Although solely addressing the SISO case, the widely diffused textbook [3, pp. 302-3041 hints that the solution of the LQRS problem can be obtained by first, finding Y = Y and Z = Z as the minimum degree solution wLr.t. Z of ( I ) only, and second, solving w.r.t. X either (2) with 2 = Z or (3) with Y = Y . The aim of this note is to study the role of (1)-(3) and conditions under which the above procedure suggested in [3] is valid. One reason for undertaking the study here reported was to better understand the nature of (1)-(3) and clarify the existing contradiction between [3] and some remarks in [4] restating the need in general of finding the minimum degree solution w.r.t. Z of ( I ) and (2). 11. RESULTS

Lemma I : A triplet of polynomial ( X , Y , Z ) is a solution of ( 1) and (2) if and only if it solves ( I ) and (3). Proof: It is already known [I], [2] that (3) is implied by (1) and (2). Then, it suffices to show that (2) is implied by ( I ) and (3). Premultiplying both sides of (3) by E , and recalling that [ I , p. 1261 EE = 1, *A2 + B2QB2, one finds 0 = (EYD,’ - B l @ ) B 2 + ( E X D , ’ - A 2 @ ) A 2

I. INTRODUCTION The monograph [ l ] shows how the discrete-time LQ stochastic regulation (LQSR) problem can be solved by polynomial equations for MIMO plants described in terms of matrix fractions. For the sake of conciseness, the reader is referred to [ 1, ch. 61 for the problem formulation, relevant notations, and definitions to which we shall adhere hereafter. Provided that the plant to be regulated [ l , Fig. 6.1.11 is free of unstable hidden modes, [ l ] proves that the transfer matrix K ( d ) E CRmp(d)of the LQ stochastic regulator

is given by

=

= (-ZB,

+ E X -A2@DI)D;‘Az

(4)

where the second equality follows from (1) and the third from the fact that [ I . p. 1311 DT1B2 = A , ’ B , D , ’ A 2 . Since D 1 and A l are both nonsingular, (2) follows from (4). 0 Lemma 2: Let B3 have full row rank. Then a triplet of polynomial matrices ( X , Y , Z ) is a solution of (1) and (2) if and only if it solves (2) and (3). Proof: Since it is already known [l], [2] that (3) is implied by (1) and (2), it suffices to show that (1) is implied by (2) and (3). Using (2) in the first line of (4),we obtain

0

K ( d ) = -M,’(d)N,(d)

+ ( E X -Az@DI)Dy‘Az

-ZA,D,‘B2

= (EYD;’

-B2U)B2 + Z B , D r 1 A ,

= (EY - B2QD2)DF1B2+ Z B , D F 1 A z

where

M l ( d ) = E - ’ ( d ) X ( d ) D , ’ ( d )E CRmm(d) N I (=~ E - ’ ( d P ’ ( d ) D F 1 ( d E)

a r n p ( 4

= ZB,

+ (EY

-

B2 *D2)Ac1B3

where the third equality follows from the fact that [ l , p. 1311 DF’ Bz = A , ’ B 3 D , ’ A 2 . Hence, (2) follows if B , has full row rank.

d and CRmp(d) denote the unit-delay operator and, respectively, the set of m x p matrices with elements in the field @ ( d )of polynomial fractions

A direct consequence of Lemmas 1 and 2 is the following. Theorem I : Under the same assumptions as in [ I , theorem (6.12)], in the indeterminate d. This solution gives rise to an asymptotically stable the solution of the LQSR problem, given in terms of the polynomial closed-loop system if and only if M , and NI above are stable-sequence matrices ( X , Y , Z ) by the minimum degree solution w.r.t. Z of (1) and matrices [ I , p. 1261. The polynomial matrices Y and X are determined, (2), can be equivalently obtained by the minimum degree solution w.r.t. together with the polynomial matrix Z ( d ) , as the minimum degree so- Z of (1) and (3) or, provided that Bj has full row rank, the minimum lution w.r.t. Z ( d ) of the following two bilateral Diophantine equations degree solution w.r.t. Z of (2) and (3). [1, p. 1271: Proof: The LSQR problem amounts to finding the minimum degree solution w.r.t. Z of ( I ) and (2). It is also known that such a solution E ( d ) Y ( d ) Z ( d ) A , ( d ) = B2(d)@Dz(d) (1) is the unique solution of (1) and (2) such that 6Z < 6 E if 6Z denotes the degree of Z . Since according to Lemma 1 (Lemma 2), the set of E ( d ) X ( d )- Z ( d ) B , ( d )= Az(d)@Di( d ) . (2) solutions ( X , Y , Z ) of (1) and (2) is the same as the set of solutions Further, any pair X ( d ) and Y ( d ) ,fulfilling (1) and (2) for some Z ( d ) , ( X , Y , Z ) of (1) and (3) [(2) and (3)], the latter has a unique element such that 6Z < 6 E coinciding with the solution of minimal degree w.r.t. satisfies Z of ( I ) and (2). The issue that is addressed hereafter is the general validity of the X ( d ) D Y i ( d ) A 2 ( d ) Y ( d ) D , ’ ( d ) B z ( d )= E ( d ) . (3) procedure that is suggested in [3]. Suppose that, as in the SISO case, a Manuscript received November 22, 1988. This work was supported by the Italian unique minimum-degree solution w.r.t. Z of (1) can be found. Let it be Department of Education and CNR. denoted by ( Y , Z ) . Is it then also (2) [(3)] solvable w.r.t. a polynomial The authors are with the Dipartimento di Sistemi e Informatica. Universita di Firenze, matrix X given Z = Z(Y = Y ) ? As the next example shows, the answer Firenze. Italy. is, in general, negative IEEE Log Number 8933219.

+

+

0018-9286/90/0300-0320$01.OO 0 1990 IEEE